1. Field of Invention
This invention relates to a method and its hardware implementation for the fundamental prediction, analysis, and parametric studies of the interaction of multiply scattered waves of any nature (acoustic, electromagnetic, elastic) with particulate composites (solid or fluid particulate phases included in a solid or fluid continuous phase). Any embodiment for the present invention constitutes a powerful research and development tool in a vast variety of engineering and scientific fields dealing with wave propagation in composites (e.g. foods, paints, cosmetics, mineral slurries, pharmaceuticals, etc). The METAMODEL™ Prediction Engine 21 , a crucial component of this invention (FIG. 1), is suitable for integration into existing acoustic or optical particle size analyzers to thereby extending their particle concentration ranges by at least two orders of magnitude.
2. Discussion of Prior Art
Prior art for the present invention pertains to three fields: a) Tools to assist in scientific and engineering work; b) Theoretical physics; and c) Instrumentation for particle size measurement. Several tools to assist in the modeling, analysis, and parametric studies of physical systems are known in the prior art and available in the market. Some of them are general mathematical tools with thousands of functions in calculus, linear algebra, ordinary differential equations, partial differential equations, Fourier analysis, dynamic systems, etc. An incomplete list includes MATLAB™ ([C1], The MathWorks, Inc), Maple V® ([C2], Waterloo Maple Software), SCIENTIFIC WORKPLACE™ ([C3], TCI Software Research), MathCad™ ([C4], MathSoft, Inc.), and Mathematica® ([C5], Wolfram Research, Inc.). These systems constitute, in a hierarchy of research and development tools, a next level up from e.g. the well-known IMSL FORTRAN/C library of scientific routines ([C6], Visual Numerics, Inc.). By becoming a little more specific in scope, they offer a higher-level language to the user, reducing the need for proficiency in computer programming and increasing the efficiency with which the scientist/engineer solves his/her problems. Some other systems are even more specific like FEMLAB® ([C7], COMSOL, Inc.) or MaX-1 ([C8], A visual Electromagnetics Platform, J. Wiley) or Real Time ([C9], A Two-Dimensional Electromagnetic Field Simulator, CRC Press). The last two systems are overly specific simulators, as they cover only certain aspects of electromagnetic waves with no flexibility offered to modeling their multiple scattering in particulates. The first one, FEMLAB®, using MATLAB™ as a software platform, offers to the user a less specific high-level language oriented towards the solution of partial differential equations using the finite element technique for problems in physics and engineering. It allows the user to create a suitable mathematical model for the physical problem at hand from available building blocks (partial differential equations, functional relationships, etc.) in an ample variety of disciplines including acoustics, fluid dynamics, structural mechanics, heat transfer, chemical processing, and electromagnetics. After a short training, the user only needs to be concerned with physics and engineering, and is required no major proficiency in computer programming and numerical intricacies. However, no pre-built modeling blocks for multiple scattering of waves in particulate composites are available either.
All these systems in the prior-art are either exceedingly general or overly specific to efficiently solve the multiply interleaved wave equations required to accurately predict the multiple scattering of waves in particulates. In order to take advantage of those existing tools, it would inevitably require an expert on both the physics and the mathematics of the problem who—in addition—would have to command the novel know-how to be disclosed in this specification, plus unacceptable computer time and human effort. A system tackling all those intricate physico-mathematical aspects of the problem, and making them transparent to the regular physicist/engineer has been clearly in need for many years. The current invention provides such a tool.
An essential part of the present invention is the ability to accurately predict the composite physical attributes (e.g. wave attenuation, wave velocity, scattered intensity, etc) when the physical attributes of the constituent phases (i.e. the particles and the host medium) are known. Two distinct approaches have historically developed in trying to mathematically model scattering of waves with particulates. One is microscopic (fundamental) and called the analytical wave-scattering theory; the other is macroscopic (phenomenological) and called the transport theory or radiative transfer theory ([22] A. Ishimaru, “Wave Propagation and Scattering in Random Media”, Academic Press, 1978; [48] F. Alba et al, “Ultrasound Spectroscopy: A Sound Approach to Sizing of Concentrated Particulates”, Handbook on Ultrasonic and Dielectric Characterization Techniques for Suspended Particulates, The American Ceramic Society, 1998). The former starts with the wave equation, rigorously predicts the scattering of a wave with a single particle in the medium, and statistically describes the interaction between all the scattered fields around all the particles, to arrive to expected values for the attributes of the composite in terms of the attributes of its constituents. It is a mathematically rigorous fundamental approach and, as a consequence, all the multiple scattering, diffraction, and interference effects are contemplated with the broad validity and accuracy range inherent to a first-principle technique. This is the physico-mathematical background of the current invention. In the latter (macroscopic) approach, instead, the basic wave equation is not the starting point. It deals directly with the transport of energy through the particulate composite, arriving to a basic differential equation (the equation of transfer). A basic assumption is that there is no correlation between the coexisting scattered fields and, therefore, their intensities can be added directly (as opposed to vectorially adding the fields). A crucial magnitude is the composite depth (particle concentration by number times total cross-section of a single particle times path length). The solution of the equation of transfer depends upon this depth and the so-called phase function. The theory is heuristic as opposed to fundamental and, ergo, it lacks the mathematical rigor and ample validity range of a first-principle approach. As it would be expected, this approach is simpler and more tractable from the numerical point of view. It has flourished during the last 100 years, in great part, due to the lack of sufficient cost-effective computational power to implement the fundamental approach. Nowadays, duplication of computer power every year or so at lower costs, has made it possible to consider full numerical implementation of the fundamental approach using non-expensive computer hardware.
Scientific literature on the analytical multiple-scattering theory is abundant but dispersed throughout a myriad of publications. Main authors who have significantly contributed in the last fifty years are L. L. Foldy, M. Lax, P. C. Waterman and R. Truell, V. Twersky, P. Lloyd and M. V. Berry, J. G. Fikioris and P. C. Waterman, A. K. Mal and S. K. Bose, N. C. Mathur and K. C. Yeh, V. N. Bringi, V. V. Varadan, V. K. Varadan, Y. Ma, L. M. Schwartz, A. J. Devaney, L. Tsang and J. A. Kong, D. D. Phanord and N. E. Berger, de Daran, F. et al, etc. Work of these researchers and others are listed in this patent. The great majority of these publications are dedicated to specific problems in specific fields of applications. A very few, however, recognizing the common underlying structural physico-mathematical core in all these fields, have attempted to formulate generic equations for the prediction of either acoustical, electromagnetic, or elastic waves in particulate composites. The publication [37] “Multiple Scattering Theory for Acoustic, Electromagnetic, and Elastic Waves in Discrete Random Media” by V.V. Varadan, V.K. Varadan and Y. Ma (Multiple Scattering of Waves in Random Media and Random Rough Surfaces, The Pennsylvania State University, 1985) belongs to this kind of general work and constitutes direct theoretical prior-art to this invention as far as the Prediction Engine 21 is concerned (FIG. 1). Even though the approach described in this prior-art and several others has a common theoretical background with the Prediction Engine 21, the latter constitutes a step forward in generality and accuracy, as the present specification will thoroughly demonstrate.
There is finally a third approach in dealing with the multiple-scattering phenomenon that is more limited in scope: it consists basically in attempting to suppress it by conceiving ad-hoc hardware designs and/or software schemes through which the multiple-scattered signal is either significantly attenuated in favor of the single-scattering component, or cross-correlation techniques are employed to filter one component from the other. U.S. Pat. Nos. 6,100,976, 5,956,139, and 4,380,817 belong to this group. This approach is mentioned here for the sake of completeness even though, since no attempt to mathematically modeling multiple-scattering phenomena is made, these patents do not truly constitute prior art to the present invention.
Two important technical fields where the present invention meets a long-term need are those of particle sizing using light scattering and acoustic spectroscopy. This is simply because, in order to indirectly measure a certain constituent attribute (e.g. particle size distribution) with a wave-based instrument, it is strictly necessary to be able to predict —as a function of this constituent attribute-that composite attribute that is directly being measured (e.g. scattering, attenuation, intensity). During the last 30 years or so, measuring of particle size in suspensions, emulsions, and aerosols using the scattering of light have developed to its mature state with a plethora of commercial instruments available in the market. Nonetheless, in practically all of those instruments a sine qua non condition for accurate performance is that the particle concentration should be reduced before analysis to extremely low values (typically under 0.1% v/v)—when actual industrial concentrations are several orders of magnitude higher. One of the main reasons for such an operating constraint is that the mathematical model (e.g. Mie theory) internally employed by these light-scattering instruments is a single-scattering model, i.e. a model that can predict accurately the composite attributes only when the particles are well-separated between each other, so that no interaction between contiguous electromagnetic fields occurs. Multiple scattering is simply not modeled.
A light-scattering instrument for aerosols (host medium is gaseous) that can work at higher concentrations (light transmissions down to 2% as opposed to typical values of about 60% to 80% in other instruments) is the one described by Harvill, T. L. and Holve, D. J. in U.S. Pat. No. 5,619,324, issued Apr. 8, 1997, in which correction for multiple scattering phenomena is conducted through a suitable adaptation of the theory described by E. D. Hirleman in [40] “Modeling of Multiple Scattering Effects in Fraunhofer Diffraction Particle Size Analysis”, Particle Characterization 5, 57-65 (1988), and [41] “A General Solution to the Inverse Near-Forward Scattering Particle Sizing Problem in Multiple Scattering Environments: Theory”, Proceedings of the 2nd International Congress on Optical Particle Sizing, Mar. 5-8 1990, pp. 159-168. With such a correction, this instrument extended the concentration range by a factor of 5-10 (only up to about a few percent by volume). Hirleman's model was developed for the specific case of small-angle scattering with the assumption that scattered fields add incoherently (on an intensity rather than amplitude basis). A “black box” approach is employed through defining a multiple-scattering redistribution matrix that represents how optical energy incident in a certain direction is redistributed by the whole composite into another direction. This matrix not only depends on the nature of medium and particles with their size distribution and concentration, but also on the geometry of the instrument as well as the optical depth of the particulate system. In plain words, the matrix can only be determined (ad hoc) once the instrument (hardware) at hand is known. Hirleman uses also a “successive orders of scattering” approach to express the multiple-scattering matrix in terms of the single-scattering one. Both matrices are global characterizations of the particulate system. No fundamental modeling of the interaction of the wave with a single particle and interaction between scattered waves around particles is undertaken. By further assuming that a scattering event of order n can be represented by the n-power of the single-scattering matrix applied to the incident signal and summing all contributions, Hirleman arrives to an analytical relation between the single and multiple-scattering matrices. The black-box approach plus the dependence of the matrix upon geometry and extension of the medium gives the model a phenomenological character as opposed to a fundamental (first principles) approach. The narrow applicability of a phenomenological approach is therefore in effect.
In U.S. Pat. No. 5,818,583, issued Oct. 6, 1998, E. Sevick-Muraca et al describe a method that applies the so-called diffusion model of light propagation to predict some aspects of multiple scattering of electromagnetic waves with suspensions (host medium is a fluid). When the particle concentration is larger than about 1%, a further approximation to the Radiative Transfer Theory can be made leading to the Diffusion Approximation Theory ([22] A. Ishimaru, 1978). In this diffusion model, it is assumed that the diffusion intensity is scattered in all directions with an almost uniform angular distribution. In what the inventors describe as photon migration techniques, the light intensity of the source is varied in time and the phase shift of the received signal for a number of different wavelengths is directly measured. The isotropic scattering coefficient is then calculated with the diffusion model and, using this coefficient, the particle size and concentration is determined inverting an integral equation that relates the particle size distribution with the scattering coefficient. Due to the diffusion regime assumption, the model cannot be used at low concentration, because the presence of substantial multiple scattering is a necessity for proper performance. Another limitation is that the referred integral equation relating the scattering coefficient with the size distribution assumes no interaction between particles occurs (the scattering coefficient is proportional to the particle concentration). This equation can only be valid when particles are far apart. As a result, the model will only be accurate when multiple scattering is strong enough for the diffusion approximation to hold but yet the particles being far apart enough so no interaction occurs between them so equation 3 in column 9, line 55 of this prior-art patent can be valid. These are opposite requirements in terms of particle concentration. Consequent with this fact, the experimental data shown in the patent correspond to concentrations around 1%—when actual industrial concentrations are much higher than that. This is specifically discussed by the authors of the patent in [46] “Photon-Migration Measurement of Latex Size Distribution in Concentrated Suspensions”, AIChE Journal, March 1997, Vol. 43, No. 3. In addition, due to the assumption of “almost uniform angular distribution of scattering”, validity of the model will break down as the ratio between the size of the particles and the wavelength increases. This is because the larger the particle size as compared to the wavelength, the stronger the directivity of the scattering in the forward direction is.
In U.S. Pat. No. 5,121,629, issued Jun. 16, 1992, a method and apparatus are described for measuring particle size distribution that employs a fundamental mathematical model for scattering of ultrasound waves in suspensions or emulsions (host medium is a fluid). The modeling aspects of this prior patent actually constitute the most direct prior-art patent to the current invention as far as its application to predicting the attenuation of acoustic waves in suspensions is concerned. Modeling details in this prior patent start in column 8, line 29 up to column 12, line 46. As in the METAMODEL™ Prediction Engine 21 of the current invention, modeling in this prior patent employs the fundamental wave scattering theory. It starts with the first-principle description of the interaction of a wave with a single particle (the particle/medium signature), and proceeds to statistically formulate the field relations within an ensemble of particles of different sizes. It ends with an analytical equation that relates the propagation constant of the composite with the propagation constant of the host medium, the physical properties of both medium and particles, as well as their size distribution and concentration. There are, however, three essential differences with the current invention when applied to acoustics. Two of them are associated with the particle/medium signature for which this prior-art patent uses the Epstein-Carhart-Allegra-Hawley (ECAH) theory while the present invention adds to this ECAH theory the following: a) modeling for possible viscoelastic behavior of both host medium and particle and b) modeling for phenomena due to surface tension which is important in gaseous particles. The third paramount difference has to do with the overlapping of contiguous fields: the prior patent arrives to equation 8 (column 10, line 60), which is the statistical relation obtained by Waterman and Truell in [9] “Multiple Scattering of Waves”, J. Math. Phys. Vol. 2, No. 4, July-August, 1961, as a closed-form dispersion equation for the particulate composite. Such a simple closed-form equation was obtained under the assumption of statistically independent point scatterers. In plain words, the expression “point scatterers” refers to the abstract concept of a particle with no physical dimensions but with the scattering properties of its actual size. It was applied in order to simplify the mathematics and its numerical coding into computer software. As a result, model accuracy deteriorates as the size of the particles and/or frequency increase. Regarding the assumption of statistical independence between scatterers positions, it eventually breaks untrue as concentration increases because of the inevitable spatial correlation between particles. Furthermore, P. Lloyd and M. V. Berry ([17] “Wave propagation through an assembly of spheres IV. Relations between different multiple scattering theories”, Proc. Phys. Soc., 1967, Vol. 91), proved there was an error in the referred Waterman and Truell's formula “arising from an artificial stratification of the medium into thin slabs”. All the referred drawbacks, though not damaging in many cases as demonstrated with the good results disclosed in this prior patent, would instead definitely contribute to destroy the accuracy of a generic model (valid for waves of any nature) if the broadest range of materials, particle size, concentration, and wave frequencies is aimed at. In fact, it will be demonstrated when the Prediction Engine 21 is described in detail that, if scatterers are treated as what they are: finite-size inclusions in the host medium, and the spatial correlation between them is not neglected, it is not possible any longer to arrive at a simple closed-form expression for the dispersion equation. Mathematics and computer programming complexities increase considerably—but prediction accuracy and range of applicability improve drastically.
In U.S. Pat. No. 6,109,098, issued Aug. 29, 2000, A. Dukhin and P. Goetz describe a device that combines acoustic and electroacoustic spectrometry to measure particle size distribution and zeta potential for concentrated dispersed systems. Focusing only on the mathematical modeling part of this prior patent, they use a macroscopic phenomenological approach to modeling the interaction between the sound wave and the concentrated suspension (host medium is a fluid). Specifically, they use the so-called “coupled phase” and “cell” models applying the long-wave regime condition, i.e. the particle size has to be much smaller than the sound wavelength. As a result, this mathematical model is only accurate for particles with sizes smaller than 10 μm. In the same patent Dukhin and Goetz state that the fundamental multiple-scattering approach “is not adequate because even the multiple scattering approach requires a single particle acoustic field which is known only for a single particle in infinite media. . . ” The present invention has overcome this difficulty recognized in the prior art conceiving a method based on the fundamental multiple-scattering approach which can work accurately in the particle size range from 1 nanometer to 1 millimeter.
In U.S. Pat. No. 6,119,510, issued Sep. 19, 2000, M. Carasso et al describe an “improved process for determining the characteristics of dispersed particles”. The authors contemplate the use of their process with acoustic as well as light waves. Focusing again only in the mathematical modeling, they describe a “computationally improved Allegra-Hawley Model” for acoustics and the original Mie Model for light waves. These two models are single-scattering models and therefore not accurate for high particle concentrations (typically over 10% for acoustics and over 1% for optics). The present invention has overcome this difficulty recognized in the prior art conceiving a method based on the fundamental multiple-scattering approach which can work accurately in the concentration range from 0.001% to 50% by volume.
In summary, there are no research and development tools available for the generic prediction, analysis, and parametric studies of the interaction between generic waves (acoustic, electromagnetic, elastic) and particulate composites. The current invention meets such a long-term need by providing a powerful research and development tool that makes all theoretical physics and numerical intricacies transparent to the human operator. A fundamental mathematical model is a key component of such a tool. The lack of a complete fundamental mathematical model for the prediction of multiple scattering of waves in particulates has, in addition, limited the concentration range of currently existing wave-based instruments. All such instruments which operation is based on the interaction of any type of wave traveling through suspensions, emulsions or solids containing inclusions, could benefit considerably from the prediction engine of this invention. Because the METAMODEL™ Prediction Engine 21 is a separate module connected to the other modules of this invention through industry-standard interfaces, it can be directly integrated to those -already existing- instruments extending their concentration ranges by several orders of magnitude. Further objects and advantages of the present invention will become apparent from a consideration of the ensuing description.